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An Extension of van Lambalgen's Theorem to Infinitely Many Relative 1-Random Reals
Miyabe, Kenshi
Notre Dame Journal of Formal Logic (2010), 51(3): 337-349
2010-07
URL
http://hdl.handle.net/2433/131806
Right
2010 © University of Notre Dame
Type
Journal Article
Textversion
publisher
Kyoto University
Notre Dame Journal of Formal Logic Volume 51, Number 3, 2010
An Extension of van Lambalgen’s Theorem to Infinitely Many Relative 1-Random Reals Kenshi Miyabe
Abstract
Van Lambalgen’s Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen’s Theorem by considering the join of infinitely many reals which are random relative to each other. In addition, we study computability of 0 the reals in the range of Omega operators. It is known that ϕ is high. We (n) extend this result to that ϕ is highn . We also prove that there exists A such A is high for some universal Turing machine M by that, for each n, the real M n using the extended van Lambalgen’s Theorem.
1 Introduction
Van Lambalgen’s Theorem provides a strong connection between randomness and computability, and it is a very powerful tool to study computability and randomness. In this paper we extend this theorem to infinitely many relative random reals. In addition, we study computability of the reals in the range of Omega operators. We use the extended van Lambalgen’s Theorem in proving that there exists a real A such A is high for some universal Turing machine M. that for each n, the real M n In Section 3 we prove two properties of martingales. This is because we extend van Lambalgen’s Theorem by martingales. One property we prove here is a saving lemma for c.e. martingales and the other is about h-order martingales. It is known that saving lemmas for computable or resource-bounded martingales hold, but the proof cannot be adapted to c.e. martingales. Here we prove a saving lemma for c.e. martingales. Again, this proof cannot be adapted to computable or resource-bounded martingales. In Section 4 we define partial strings and expand the domain of martingales from strings to the partial strings. By using these extended martingales we can strengthen
Received November 30, 2009; accepted December 8, 2009; printed June 16, 2010 2010 Mathematics Subject Classification: Primary, 03D32; Secondary, 03D25 Keywords: van Lambalgen’s Theorem, martingale, high, Omega operator c 2010 by University of Notre Dame 10.1215/00294527-2010-020
337
338
Kenshi Miyabe
the saving lemma. In Section 5 we study van Lambalgen’s Theorem. Van Lambalgen’s Theorem is about two relative random reals and can deal with finitely many relative random reals. We extend this theorem to infinitely many relative random reals. In Section 6 we prove some results about the computability of the reals in the 0 (n) range of Omega operators. It is known that ϕ is high. We extend this to that ϕ A = ϕ (n) is highn . We also prove that there exists A such that, for each n, the real M for some universal prefix-free Turing machine M. 2 Preliminaries
Now we look at notations we use in this paper and basic definitions. For a more complete introduction, see Soare [13] or Odifreddi [10; 11] for computability theory and Li and Vitányi [7], Downy and Hirschfeldt [3], or Nies [9] for algorithmic randomness. We say that ψ is a partial computable function from Nk to N if there is a Turing machine P such that ψ(x0 , . . . , xk−1 ) = y if and only if P on inputs x0 , . . . , xk−1 outputs y. For a set A of natural numbers, the set A0 = {e | 8eA (e)} is called the jump of A where 8eA is the eth partial computable function N → N with A as an oracle. We write A(n) to mean nth jump of A. We say that A is T-reducible to B, written as A ≤T B, if A = 8eB for some e. We can regard a set A as an infinite binary sequence such that the ith bit of the sequence is 1 if i ∈ A and 0 if i 6 ∈ A. The Cantor space, denoted by 2ω , is the set of all infinite binary sequences and 2
Author(s)
Citation
Issue Date
An Extension of van Lambalgen's Theorem to Infinitely Many Relative 1-Random Reals
Miyabe, Kenshi
Notre Dame Journal of Formal Logic (2010), 51(3): 337-349
2010-07
URL
http://hdl.handle.net/2433/131806
Right
2010 © University of Notre Dame
Type
Journal Article
Textversion
publisher
Kyoto University
Notre Dame Journal of Formal Logic Volume 51, Number 3, 2010
An Extension of van Lambalgen’s Theorem to Infinitely Many Relative 1-Random Reals Kenshi Miyabe
Abstract
Van Lambalgen’s Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen’s Theorem by considering the join of infinitely many reals which are random relative to each other. In addition, we study computability of 0 the reals in the range of Omega operators. It is known that ϕ is high. We (n) extend this result to that ϕ is highn . We also prove that there exists A such A is high for some universal Turing machine M by that, for each n, the real M n using the extended van Lambalgen’s Theorem.
1 Introduction
Van Lambalgen’s Theorem provides a strong connection between randomness and computability, and it is a very powerful tool to study computability and randomness. In this paper we extend this theorem to infinitely many relative random reals. In addition, we study computability of the reals in the range of Omega operators. We use the extended van Lambalgen’s Theorem in proving that there exists a real A such A is high for some universal Turing machine M. that for each n, the real M n In Section 3 we prove two properties of martingales. This is because we extend van Lambalgen’s Theorem by martingales. One property we prove here is a saving lemma for c.e. martingales and the other is about h-order martingales. It is known that saving lemmas for computable or resource-bounded martingales hold, but the proof cannot be adapted to c.e. martingales. Here we prove a saving lemma for c.e. martingales. Again, this proof cannot be adapted to computable or resource-bounded martingales. In Section 4 we define partial strings and expand the domain of martingales from strings to the partial strings. By using these extended martingales we can strengthen
Received November 30, 2009; accepted December 8, 2009; printed June 16, 2010 2010 Mathematics Subject Classification: Primary, 03D32; Secondary, 03D25 Keywords: van Lambalgen’s Theorem, martingale, high, Omega operator c 2010 by University of Notre Dame 10.1215/00294527-2010-020
337
338
Kenshi Miyabe
the saving lemma. In Section 5 we study van Lambalgen’s Theorem. Van Lambalgen’s Theorem is about two relative random reals and can deal with finitely many relative random reals. We extend this theorem to infinitely many relative random reals. In Section 6 we prove some results about the computability of the reals in the 0 (n) range of Omega operators. It is known that ϕ is high. We extend this to that ϕ A = ϕ (n) is highn . We also prove that there exists A such that, for each n, the real M for some universal prefix-free Turing machine M. 2 Preliminaries
Now we look at notations we use in this paper and basic definitions. For a more complete introduction, see Soare [13] or Odifreddi [10; 11] for computability theory and Li and Vitányi [7], Downy and Hirschfeldt [3], or Nies [9] for algorithmic randomness. We say that ψ is a partial computable function from Nk to N if there is a Turing machine P such that ψ(x0 , . . . , xk−1 ) = y if and only if P on inputs x0 , . . . , xk−1 outputs y. For a set A of natural numbers, the set A0 = {e | 8eA (e)} is called the jump of A where 8eA is the eth partial computable function N → N with A as an oracle. We write A(n) to mean nth jump of A. We say that A is T-reducible to B, written as A ≤T B, if A = 8eB for some e. We can regard a set A as an infinite binary sequence such that the ith bit of the sequence is 1 if i ∈ A and 0 if i 6 ∈ A. The Cantor space, denoted by 2ω , is the set of all infinite binary sequences and 2
